Dominant (temperament): Difference between revisions
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{{Infobox | {{Infobox regtemp | ||
| Title = Dominant | | Title = Dominant | ||
| Subgroups = 2.3.5.7 | | Subgroups = 2.3.5.7 | ||
| Comma basis = [[36/35]], [[64/63]] | | Comma basis = [[36/35]], [[64/63]] | ||
| Edo join 1 = | | Edo join 1 = 12 | Edo join 2 = 17c | ||
| | | Generators = 3/2 | Generators tuning = 701.1 | Optimization method = CWE | ||
| MOS scales = [[2L 3s]], [[5L 2s]], [[5L 7s]] | | MOS scales = [[2L 3s]], [[5L 2s]], [[5L 7s]] | ||
| Mapping = 1; 1 4 -2 | | Mapping = 1; 1 4 -2 | ||
| Line 11: | Line 11: | ||
| Odd limit 2 = (7-limit) 27 | Mistuning 2 = 26.3 | Complexity 2 = 12 | | Odd limit 2 = (7-limit) 27 | Mistuning 2 = 26.3 | Complexity 2 = 12 | ||
}} | }} | ||
'''Dominant''' is a [[regular temperament|temperament]] which is an [[extension]] of both [[meantone]] and [[archy]]. It is defined by [[tempering out]] the [[81/80|syntonic comma (81/80)]] and [[64/63|septimal comma (64/63)]] in the 7-limit. It also tempers out the [[36/35|septimal quartertone (36/35)]], as 36/35 = (64/63)(81/80). It is the unique temperament that identifies the [[harmonic seventh chord]] with the [[dominant seventh chord]], which is a familiar feature from [[12edo]]. | '''Dominant''' is a [[regular temperament|temperament]] which is an [[extension]] of both [[meantone]] and [[archy]]. It is defined by [[tempering out]] the [[81/80|syntonic comma (81/80)]] and [[64/63|septimal comma (64/63)]] in the 7-limit. It also tempers out the [[36/35|septimal quartertone (36/35)]], as 36/35 = (64/63)⋅(81/80). It is the unique temperament that identifies the [[harmonic seventh chord]] with the [[dominant seventh chord]], which is a familiar feature from [[12edo]]. | ||
However, it is not very accurate for the same reason that 12edo is inaccurate in the 7-limit, as either 5/4 or 7/4 must be tuned very sharply (with 5/4 reaching over 462 cents in the best tuning of 7/4, and likewise 7/4 reaching over 1006 cents in the best tuning of 5/4). Thus, the | However, it is not very accurate for the same reason that 12edo is inaccurate in the 7-limit, as either 5/4 or 7/4 must be tuned very sharply (with 5/4 reaching over 462 cents in the best tuning of 7/4, and likewise 7/4 reaching over 1006 cents in the best tuning of 5/4). Thus, the best tuning is a compromise between the two, tuning 3/2 basically just. | ||
The most obvious extension to the 11 and 13-limit is treating the major and minor thirds as [[14/11]] and [[13/11]] as well as 5/4 and 6/5, tempering out [[56/55]] and [[66/65]]. This favors even sharper fifths on the edge of the [[gentle region]]. 29edo tunes this about as well as possible, albeit using the second best approximation of most harmonics. | The most obvious extension to the 11- and 13-limit is treating the major and minor thirds as [[14/11]] and [[13/11]] as well as 5/4 and 6/5, tempering out [[56/55]] and [[66/65]]. This favors even sharper fifths on the edge of the [[gentle region]]. [[29edo]] tunes this about as well as possible, albeit using the second best approximation of most harmonics. | ||
Other possible tunings include [[17edo]] (17c val), [[41edo]] (41cd val), [[53edo]] (53cdd val), as well as [[Pythagorean tuning]]. | Other possible tunings include [[17edo]] (17c val), [[41edo]] (41cd val), [[53edo]] (53cdd val), as well as [[Pythagorean tuning]]. | ||
| Line 46: | Line 46: | ||
| 7 || 107.8 || 15/14 | | 7 || 107.8 || 15/14 | ||
|} | |} | ||
<nowiki />* In 7-limit [[CWE]] tuning | <nowiki/>* In 7-limit [[CWE]] tuning | ||
== Chords and harmony == | == Chords and harmony == | ||
Much of 12edo harmony can be used. Dominant enables chords of [[didymic chords|didymic]] and [[archytas chords|archytas]]. | Much of 12edo harmony can be used. Dominant enables chords of [[didymic chords|didymic]] and [[archytas chords|archytas]]. The dominant seventh chord represents the harmonic seventh chord, whereas the [[German sixth chord|(German) augmented sixth chord]] is more or less equivalent to meantone's dominant seventh chord, as a tuning of [[28:35:42:50|1–5/4–3/2–25/14]]. | ||
== Tunings == | == Tunings == | ||
| Line 69: | Line 69: | ||
=== Tuning spectrum === | === Tuning spectrum === | ||
{{todo|complete table|comment=add the missing 11- and 13-limit eigenmonzo tunings}} | |||
{| class="wikitable center-all left-4" | {| class="wikitable center-all left-4" | ||
|- | |- | ||
| Line 96: | Line 97: | ||
| [[1/5-comma meantone|1/5 syntonic comma]] | | [[1/5-comma meantone|1/5 syntonic comma]] | ||
|- | |- | ||
| [[12edo|7\12]] | | '''[[12edo|7\12]]''' | ||
| | | | ||
| 700.000 | | '''700.000''' | ||
| Lower bound of 7- and 9-odd-limit diamond monotone | | '''Lower bound of 7- and 9-odd-limit diamond monotone''' | ||
|- | |- | ||
| | | | ||
| Line 114: | Line 115: | ||
| [[7/5]] | | [[7/5]] | ||
| 702.915 | | 702.915 | ||
| 7 & 9-odd limit minimax tuning | | 7- & 9-odd-limit minimax tuning | ||
|- | |- | ||
| | | | ||
| Line 156: | Line 157: | ||
| 1/2 septimal comma | | 1/2 septimal comma | ||
|- | |- | ||
| [[5edo|3\5]] | | '''[[5edo|3\5]]''' | ||
| | | | ||
| 720.000 | | '''720.000''' | ||
| Upper bound of 7- and 9-odd-limit diamond monotone | | '''Upper bound of 7- and 9-odd-limit diamond monotone''' | ||
|- | |- | ||
| | | | ||
| Line 166: | Line 167: | ||
| Full septimal comma | | Full septimal comma | ||
|} | |} | ||
<nowiki />* Besides the octave | <nowiki/>* Besides the octave | ||
== References == | == References == | ||